Wide-angle exclusive scattering - an update

Nuclear Physics A 782 (2007) 77c–85c

Wide-angle exclusive scattering - an update P. Kroll Fachbereich Physik, Universit¨at Wuppertal, D-42097 Wuppertal, Germany The handbag mechanism for wide-angle exclusive scattering is discussed and compared with other theoretical approaches. Its application to Compton scattering, meson photoproduction and two-photon annihilations into pairs of hadrons is reviewed in some detail. 1. Introduction Recently a new approach to wide-angle Compton scattering off protons has been proposed [1,2] where, for Mandelstam variables s, −t, −u that are large as compared to a typical hadronic scale, Λ2 of the order of 1 GeV2 , the process amplitudes factorize into a hard parton-level subprocess, Compton scattering off quarks, and in soft form factors which represent 1/x moments of generalized parton distributions (GPDs) and encode the soft physics (see Fig. 1a ). Subsequently it has been realized that this so-called handbag mechanism also applies to a number of other wide-angle reactions such as virtual Compton scattering [3] (provided the photon virtuality, Q2 is smaller than −t), meson photoand electroproduction [4] or two-photon annihilations into pairs of mesons [5] or baryons [6,7] (see Fig. 1b). Last not least the photon-pseudoscalar meson transition form factor is to be mentioned (see Fig. 1c) which is controlled by the same subprocess, γ ∗ γ (∗) → q q¯ as the other two-photon annihilation processes. It should be noted that the handbag mechanism bears resemblance to the treatment of inelastic Compton scattering advocated for by Bjorken and Paschos [8] long time ago. There are other mechanisms which may also contribute to wide-angle scattering besides the handbag which is characterized by one active parton, i.e. one parton from each hadron participates in the hard subprocess (e.g. γq → γq in Compton scattering) while all others are spectators. On the one hand, there are the so-called cat’s ears graphs (see Fig. 1d) with two active partons participating in the subprocess (e.g. γqq → γqq). It can be shown however that in these graphs either a large parton virtuality or a large parton transverse momentum occurs. This forces the exchange of at least one hard gluon. Hence, the cat’s ears contribution is expected to be suppressed as compared to the handbag one. The next class of graphs are characterized by three active quarks (e.g. γqqq → γqqq) which require the exchange of at least two hard gluons (see Fig. 1e). One can go on and consider four active partons and so forth. The series generated that way, bears resemblance to an expansion in terms of n-body operators used in many-body theory. In principle, all the different contributions have to be added coherently. In practice, however, this is a difficult, currently almost impossible task since each contribution has its own associated soft hadronic matrix element which, as yet, cannot be calculated from QCD and is in general phenomenologically unknown. 0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.10.005

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γ (∗)

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Figure 1. Upper panel from left to right: Handbag diagram for Compton scattering (a), two-photon annihilation into a pair of hadrons (b) and the γP transition form factor (c). Lower panel: Cat’s ears (d), three-particle correlation (e) and the ERBL contribution (f).

p

Let me return to the case of three active partons. In valence quark approximation there is no spectator left. Hence, the GPD blob decays into two hadron distribution amplitudes and one arrives at the ERBL factorization scheme [9] (see Fig. 1f). This contribution is expected to dominate at asymptotically large momentum transfer; the handbag contribution is formally a power correction to it. However, for momentum transfer of the order of 10 GeV2 , characteristic of current wide-angle experiments, the ERBL contribution is about a factor of 10−2 − 10−3 below experiments for reactions involving baryons. The onset of the ERBL region is expected to occur above −t(−u) > 100 GeV2 . A property of the ERBL factorization scheme are the dimensional counting rules which say that for asymptotically large scales where all soft scales can be ignored, exclusive observables behave proportional to certain powers of the hard scale. The powers are determined by the number of point-like particles participating in the hard scattering. Perturbative QCD modifies these power behaviour by logs arising from the running coupling constant and the evolution of the soft hadronic matrix elements. As yet no evidence for these logs has been observed in experiment. In view of this and the ununderstood normalization it is premature to claim dominance of the ERBL contribution from the occasionally observed success of dimensional counting. Moreover, recent precise experiments reveal clear deviations from dimensional counting. Thus, for instance the Compton cross section, measured recently at JLab [10], should scale as s−6 at fixed c.m.s. scattering angle but it rather behaves as s−7...8, see Fig. 2. The Pauli form factor should fall down as 1/t3 but, pas a matter of fact, it behaves as 1/t2 . The integrated cross section for γγ → p¯ p(π + π − ) drops with the power ≃ 7.2(4) instead of 5(3) [11,12]. The handbag mechanism in real Compton scattering is reviewed in some detail in Sect. 2. The large −t behaviour of the GPDs and their associated form factors is discussed in Sect. 3 and predictions for Compton scattering are given. A few results for wide-angle meson photoproduction and two-photon annihilations into pairs of hadrons are presented 0

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Figure 2. Left: The Compton cross section, scaled by s6 versus s at a c.m.s. scattering angle of 90◦ . Data are taken from [10]. Right: The Compton form factors RV , RA and RT scaled by t2 . in Sect. 4 and 5, respectively. The paper ends with a summary (Sect. 6). 2. Wide-angle Compton scattering For Mandelstam variables s, −t and −u that are large as compared to a typical hadronic scale Λ2 where Λ being of order 1 GeV, it can be shown that the handbag diagram shown in Fig. 1a, is of relevance in wide-angle Compton scattering. To see this it is of advantage to work in a symmetrical frame which is a c.m.s rotated in such a way that the momenta of the incoming (p) and outgoing (p′ ) proton momenta have the same light-cone plus components. In this frame skewness, defined as ξ =

(p − p′ )+ , (p + p′ )+

(1)

is zero. The bubble in the handbag is viewed as a sum over all possible parton configurations as in deep ineleastic lepton-proton scattering. The crucial assumptions in the handbag approach are that of restricted parton virtualities, ki2 < Λ2 , and of intrinsic transverse parton momenta, k⊥i , defined with respect to their parent hadron’s momen2 /xi < Λ2 , where xi is the momentum fraction parton i carries. One tum, which satisfy k⊥i can then show [2] that the subprocess Mandelstam variables sˆ and uˆ are the same as the ones for the full process, Compton scattering off protons, up to corrections of order Λ2 /t: sˆ = (kj + q)2 ≃ (p + q)2 = s ,

uˆ = (kj − q ′ )2 ≃ (p − q ′ )2 = u .

(2)

The active partons, i.e. the ones to which the photons couple, are approximately on-shell, move collinear with their parent hadrons and carry a momentum fraction close to unity, xj , x′j ≃ 1. Thus, like in deep virtual Compton scattering, the physical situation is that of a hard parton-level subprocess, γq → γq, and a soft emission and reabsorption of quarks from the proton.

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The handbag approach leads to the following result for the Compton cross section dˆ σ dσ = dt dt



RV (T ) (t) =

X

−t 2 1 2 2 [RV (t) + R (t) + RA (t)] 2 4m2 T  −t 2 us 2 2 [R (t) + R (t) − R (t)] + O(αs ) , (3) − 2 V T A s + u2 4m2 where dˆ σ /dt is the Klein-Nishina cross section for Compton scattering off massless, pointlike spin-1/2 particles of charge unity. The hard scattering has been calculated to next-toleading order (NLO) perturbative QCD [13]. It turned out that the NLO amplitudes are ultraviolet regular but those amplitudes which are non-zero to LO, are infrared divergent. As usual the infrared divergent pieces are interpreted as non-perturbative physics and absorbed into the soft form factors, Ri . Thus, factorization of the wide-angle Comton amplitudes within the handbag approach is justified to (at least) NLO. To this order the gluonic subprocess, γg → γg, has to be taken into account as well which goes along with corresponding gluonic GPDs and their associated form factors. In the numerical results on Compton scattering the NLO corrections are included. The form factors in Eq. (3) are specific to Compton scattering. They represent 1/¯ x f and E moments of the GPDs H, H e2a

a

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RA (t) =

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d¯ x fa (¯ sign(¯ x) H x, 0; t) . x¯

(4)

Determinations of the zero-skewness GPDs from nucleon form factor data have been attempted recently [14,15]. The nucleon form factors only provide the first moment of the GPDs, e.g. F1 (t) =

X a

ea

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1

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(5)

−1

Hence, a determination of the GPDs is only possible with the help of a parameterization. The following ansatz is used in Ref. [14]: x, 0, t) = q a (¯ x) exp [fa (¯ x)t] , H a (¯

(6)

where q a is the usual parton distribution for quarks of flavor a and the profile function fa combines properties of the Regge pole model at low x¯ with light-cone wavefunction overlaps at large x¯. The parton distributions are taken from CTEQ [16]. Fits to the data on the Dirac and Pauli form factors for proton and neutron as well on the axialvector form factor yield the GPDs for valence quarks. The Compton form factors evaluated from them, are shown in Fig. 2. With the Compton form factors at hand various Compton observables can be calculated in a parameter-free way. The predictions for the differential cross section are shown in Fig. 3 and compared to the Jlab data [10]. For sufficiently large −t and −u, actually for −t, −u > 2.5 GeV2 , fair agreement with experiment is to be seen. Another interesting observable is the helicity correlation, ALL , between the initial state photon and proton or, equivalently, the helicity transfer, KLL , from the incoming photon to the outgoing proton. In the handbag approach one obtains [3,13] RA + O(κT , αs ) , ALL = KLL ≃ AˆLL RV

(7)

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Figure 3. Handbag predicitions for the Compton cross section at s = 6.79, 8.9, 10.92 and 20 GeV2 (left) and for the helicity correlations ALL = KLL at s = 11 GeV2 (right). Data are taken form Refs. [10,17]. The green bands represent the uncertainties of the handbag predictions due to the errors of the Compton form factors. The yellow bands are estimates of the target mass corrections [18]. where AˆLL = (s2 − u2 )/(s2 + u2 ) is the corresponding observable for γq → γq. The result (7) is a robust prediction of the handbag mechanism, the magnitude of the subprocess helicity correlation is only slightly diluted by the ratio of the form factors RA and RV . Predictions for ALL = KLL are shown in Fig. 3. It is to be emphasized that the handbag prediction for the helicity correlation is only mildly energy dependent (the dashed line in Fig. 3 represents ALL at s = 20 GeV2 ). The JLab E99-114 collaboration [17] has measured KLL at a c.m.s. scattering angle of 120◦ and a photon energy of 3.23 GeV. Although the kinematical requirement of large Mandelstam variables is not satisfied for this measurement - and in so far one has to reckon with large dynamical and kinematical corrections - fair agreement with the handbag predicition is to be observed. Calculations on the basis of the ERBL factorization scheme, e.g. Ref. [19], reveal severe difficulties in getting the normalization of the Compton cross section correctly, the numerical results are way below experiment. There is growing evidence [20] 1 that the proton’s distribution amplitude is close to the asymtotic form ∝ x1 x2 x3 . Evaluating the Compton cross section from such a distribution amplitude one obtains results that are too small by a factor of about 10−3 . Moreover, the ERBL approach [19] leads to a negative value for KLL at angles larger than 90◦ in conflict with the JLab result [17]. Thus, one is forced to conclude that wide-angle Compton scattering at energies available at JLab is not dominated by the ERBL contribution. Fairly good agreement with the Jlab results is, on the other hand, obtained with a constituent quark model [22]. 1

A perturbatively calculated J/Ψ → p¯ p decay width only agrees with experiment if a proton distribution amplitude close to the asymptotic form is employed [21]

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3. Two-photon annihilations into pairs of hadrons The arguments for handbag factorization hold as well for two-photon annihilations into pairs of hadrons as has recently been shown in Ref. [5] (see also Ref. [7]). The cross section for γγ ↔ pp reads 2 o 4πα2 n γ dσ ( γγ ↔ pp ) = 2 elm |Reff (s)|2 + cos2 θ |RVγ (s)|2 , 2 dt s sin θ r s γ γ |RA + RPp |2 + |Rγp | . Reff = 4m2

(8)

In analogy to Eq. (4) the form factors represent moments of two-hadron distribution amplitudes (TDAs), Φ2h , which are time-like versions of GPDs. Since the TDAs are unknown at present one has to extract the form factors from experiment. A fit [23] to the recent BELLE data [11] is shown in Fig. 4. In combination with the data on the integrated cross section for s ≥ 8 GeV2 one finds for the form factors the parameterizations s2 RVγ = (8.20 ± 0.77) GeV4 (s/s0 )(−1.10±0.15) , γ s2 Reff = (2.90 ± 0.31) GeV4 (s/s0 )(−1.10±0.15) ,

(9)

where s0 = 10.4 GeV2 . The energy dependence of the scaled form factors manifests the violation of dimensional counting. The time reversed process pp → γγ may be measured at the future FAIR facility. One may measure the cross section and the form factors at higher energies, test factorization and, in combination with data on the electromagnetic form factors GM , F2 for protons and neutrons in the time-like region, one may attempt an analysis of the TDAs. Measurement of the correlation between the proton and antiproton helicities will allow to disentangle the axial vector and the pseudoscalar form factors. The cross section (8) can be straightforwardedly extended to other baryons. Relying on flavor symmetry one can evaluate the cross sections for two-photon annihilations into ΛΛ and 0 Σ0 Σ from the form factors (9). Reasonable agreement with experiment is found [6]. It is to be stressed that the ERBL mechanism has again difficulties to account for the size of the cross sections [24] while the diquark model [25] which is a variant of the ERBL approach in which diquarks are considered as quasi-elementary constituents of baryons, ¯ is infair agreement with experiment for γγ → B B. Two-photon annihilations into meson pairs can also be investigated within the handbag approach. For instance for pseudoscalar mesons the cross section reads 8πα2 dσ (γγ → MM ) = 2 elm |R (s)|2 , dt s sin4 θ M M

(10)

The characteristic 1/ sin4 θ dependence is in good agreement with the BELLE data [12] 2 on the cross sections for pion and Kaon pairs for s > ∼ 9 GeV , see Fig. 4. A feature of the handbag mechanism in the time-like region is the intermediate qq state implying the absence of isospin-two components in the final state. A consequence of this property is dσ dσ (γγ → π 0 π 0 ) = (γγ → π + π − ) , dt dt

(11)

2 Note that in the time-like region the pseudoscalar form factor contributes while it decouples in the space-like region.

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Figure 4. Handbag predictions for the differential cross sections for γγ → pp (left) and π + π − (•), K = K − (◦) (right). Data are taken form Refs. [11,12]. which is independent of the soft physics input and is, in so far, a robust prediction of the handbag approach. This result also holds for ρ mesons. The ERBL approach leads to substantial differences between the cross sections for charged and uncharged meson pairs. Measurements of these cross sections are highly welcome. Neglecting non-valence quark form factors one obtains for Kaon pairs 2 dσ dσ (γγ → KS KS ) = (γγ → K + K − ) , dt 25 dt

(12)

in the strict flavor symmetry limit. Preliminary BELLE results [26] on the KS KS cross section seem to indicate that the experimental ratio of these two cross sections is below that limit. Evaluating the γγ → MM cross section within the ERBL approach from DAs that are close to the asymptotic form one obtains results that are nearly an order of magnitude below experiment even if the large NLO corrections are taken into account [27]. The differential cross section in this approach is approximately similar to the handbag result (10) with the ERBL result for the pion’s electromagnetic form factor instead of the annihilation one. Brodsky and Lepage [28] recommended the use of the experimental form factor instead of the ERBL result. A recent CLEO measurement [29] of Fπ however refute this idea. 4. Pion photoproduction and pp → γπ 0 Photo- and electroproduction of mesons as well as the s ↔ t crossed reaction, protonantiproton annihilation into photon and meson have also been analyzed within the handbag approach [4,23,30]. Considering the case of the π 0 as an example, one obtains the following cross sections "

#

 0 2 −t  π0 2 t2 t(s − u)2  π0 2 dσ π (γp → π 0 p) = αelm a2 R (t) + R (t) + RA (t) , V T dt 32s4 u2 4m2 (s − u)2

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 0 2  dσ |a|2 π0 (pp → γπ 0 ) = αelm 3 4 Reff (s)|2 + cos2 θ RVπ (s) . dt s sin θ

(13)

Due to the universality of the GPDs/TDAs the new form factors can be either evaluated from the GPDs determined in Ref. [14] or estimated from the annihilation form factors (9) assuming a constant ratio of the form factors for d and u quarks with values lying in the range 0.25 − 0.75. For asymptotically large Mandelstam variables the meson is generated by one-gluon exchange leading to 16 αs fπ h1/τ iπ , (14) 9 where fπ is the pion’s decay constant and h1/τ iπ the 1/τ moment of its DA. This result is too small to account for the experimental values of the cross sections. Dynamical effects as for instance higher orders of perturbative QCD, transverse degrees of freedom or power corrections, are required in order to built up the proper values of a and a. A quantitative estimates of these constants are however lacking at present. Taking them as free parameters a fair description of the photoproduction [31] and pp → γπ 0 [32] data is obtained. Interestingly the helicity correlation parameter ALL = KLL for photoproduction is similar to that for Compton scattering. Also the ratio of π + and π − photoproduction a = a =

"

dσ(γn → π − p) ed uˆ + eu sˆ ≃ + dσ(γp → π n) eu uˆ + ed sˆ

#2

,

(15)

is in fair agreement with a JLab measurement [33]. The latter two results are independent on the cross section normalization. 5. Summary I briefly reviewed the theoretical activities on applications of the handbag mechanism to wide-angle scattering. There are many interesting predictions, some are in fair agreement with experiment, others still awaiting their experimental examination. It seems that the handbag mechanism plays an important role in exclusive scattering for momentum transfers of the order of 10 GeV2 . However, before we can draw firm conclusions more experimental tests are needed. The ERBL approach, on the other hand, typically provides cross sections which are way below experiment. I finally emphasize that the structure of the handbag amplitude, namely its representation as a product of perturbatively calculable hard scattering amplitudes and t-dependent form factors is the essential result. Refuting the handbag approach necessitates experimental evidence against this factorization. REFERENCES 1. A. V. Radyushkin, Phys. Rev. D 58, 114008 (1998) [hep-ph/9803316]. 2. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Eur. Phys. J. C 8, 409 (1999) [hepph/9811253]. 3. M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Phys. Lett. B 460, 204 (1999) [hepph/9903268].

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